Basically you use a double-precision floating point number for the real part, a double-precision floating point number for the imaginary part, and write methods for any operation you want to include (such as addition, etc.; trigonometric functions, exponential function).
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Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.
The complex numbers are a field.
All of them. Real numbers are a subset of complex numbers.
Gerolamo Cardano is an Italian mathematician who introduced complex numbers. Complex numbers are those that can be expressed in the form of a+bi where a and b represent real numbers.
Yes, all real numbers are complex numbers.